scholarly journals Softening the Complexity of Entropic Motion on Curved Statistical Manifolds

2012 ◽  
Vol 19 (01) ◽  
pp. 1250001 ◽  
Author(s):  
Carlo Cafaro ◽  
Adom Giffin ◽  
Cosmo Lupo ◽  
Stefano Mancini
Keyword(s):  
Statistical Model ◽  
Uncertainty Relation ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Temporal Complexity ◽  
Additional Information ◽  
Statistical Manifolds ◽  
Higher Dimensional ◽  
Jacobi Vector ◽  
Minimum Uncertainty

We study the information geometry and the entropic dynamics of a three-dimensional Gaussian statistical model. We then compare our analysis to that of a two-dimensional Gaussian statistical model obtained from the higher-dimensional model via introduction of an additional information constraint that resembles the quantum mechanical canonical minimum uncertainty relation. We show that the chaoticity (temporal complexity) of the two-dimensional Gaussian statistical model, quantified by means of the information geometric entropy (IGE) and the Jacobi vector field intensity, is softened with respect to the chaoticity of the three-dimensional Gaussian statistical model.

Asymptotic Fr echet ´Differentiability

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Tiˇser Jaroslav
Keyword(s):  
Hilbert Spaces ◽  
Variational Approach ◽  
Three Dimensional ◽  
Mean Value ◽  
Two Dimensional ◽  
Current Development ◽  
Lipschitz Mappings ◽  
Higher Dimensional ◽  
Fréchet Differentiability ◽  
Frechet Differentiability

This chapter presents the current development of the first, unpublished proof of existence of points Fréchet differentiability of Lipschitz mappings to two-dimensional spaces. For functions into higher dimensional spaces the method does not lead to a point of Gâteaux differentiability but constructs points of asymptotic Fréchet differentiability. The proof uses perturbations that are not additive, rather than the variational approach, but still provides (asymptotic) Fréchet derivatives in every slice of Gâteaux derivatives. However, it cannot be used to prove existence of points of Fréchet differentiability of Lipschitz mappings of Hilbert spaces to three-dimensional spaces. The results are negative in the sense that an appropriate version of the multidimensional mean value estimate holds.

scholarly journals Three-Dimensional Volume-Rendered Series Complements 2D Orthogonal Multidetector Computed Tomography in the Evaluation of Abnormal Spinal Curvature in Patients at a Major Cancer Center: A Retrospective Review

2012 ◽  
Vol 2012 ◽  
pp. 1-6 ◽  
Author(s):  
J. Matthew Debnam ◽  
Leena Ketonen ◽  
Nandita Guha-Thakurta
Keyword(s):  
Computed Tomography ◽  
Retrospective Review ◽  
Three Dimensional ◽  
Spinal Curvature ◽  
Cancer Center ◽  
Two Dimensional ◽  
Plain Radiographs ◽  
Additional Information ◽  
Image Set ◽  
Mass Lesions

Background. Abnormal spinal curvature is routinely assessed with plain radiographs, MDCT, and MRI. MDCT can provide two-dimensional (2-D) orthogonal as well as reconstructed three-dimensional volume-rendered (3-D VR) images of the spine, including the translucent display: a computer-generated image set that enables the visualization of surgical instrumentation through bony structures. We hypothesized that the 3-D VR series provides additional information beyond that of 2-D orthogonal MDCT in the evaluation of abnormal spinal curvature in patients evaluated at a major cancer center. Methods. The 3-D VR series, including the translucent display, was compared to 2-D orthogonal MDCT studies in patients with an abnormal spinal curvature greater than 25 degrees and scored as being not helpful (0) or helpful (1) in 3 categories: spinal curvature; bony definition; additional findings (mass lesions, fractures, and instrumentation). Results. In 38 of 48 (79.2%) patients assessed, the 3-D VR series were scored as helpful in 63 of 144 (43.8%) total possible categories (32 spinal curvature; 14 bony definition; 17 additional findings). Conclusion. Three-dimensional MDCT images, including the translucent display, are complementary to multiplanar 2-D orthogonal MCDT in the evaluation of abnormal spinal curvature in patients treated at a major cancer center.

An Interferometer Based Experimental Technique to Evaluate Large Strains and Springback on Sheet Metal

2013 ◽  
Author(s):  
Luis Rafael Sanchez ◽  
Shannon Peterson ◽  
Carl G. Simonsen ◽  
Abrar Satar
Keyword(s):  
Sheet Metal ◽  
Three Dimensional ◽  
Bending Moment ◽  
Large Strains ◽  
Two Dimensional ◽  
Strain Gages ◽  
Additional Information ◽  
Software Algorithms ◽  
Dimensional Measurements ◽  
Vertical Scanning Interferometry

A technique was successfully developed to measure large tensile, compressive strains, springback and strain reversal effects on sheet metal bent to small radii. Vertical Scanning Interferometry (VSI) was used to measure three dimensional data from surfaces with sides varying from 160 nm to 2 mm. Software algorithms were utilized to determine surface topography maps from three-dimensional curved locations and to represent them in a two dimensional plane. Fine reference marks were engraved on both sides of sample. The sample was bent /unbent to small radii under a pure bending moment. Outer strains were calculated from VSI two-dimensional measurements of the original and final lengths between the reference marks. Strain gages, applied at locations close to the reference marks, gave additional information at the elasto-plastic range. Experimental data collected included bending moment as a function of strain, 3-D curvature profiles, springback and reverse bending effects. The technique was proved useful for the experimental evaluation and theoretical validation of bending and springback properties of sheet metal. Experimental results for aluminum and steel alloys are presented.

scholarly journals ON THE SYMMETRIES OF INTEGRABILITY

1992 ◽  
Vol 06 (11n12) ◽  
pp. 1881-1903 ◽  
Author(s):  
M. BELLON ◽  
J-M. MAILLARD ◽  
C. VIALLET
Keyword(s):  
Quantum Groups ◽  
Discrete Group ◽  
Three Dimensional ◽  
Algebraic Combinatorics ◽  
Two Dimensional ◽  
Algebraic Varieties ◽  
Spin Models ◽  
Projective Transformations ◽  
Vertex Models ◽  
Higher Dimensional

We show that the Yang-Baxter equations for two-dimensional models admit as a group of symmetry the infinite discrete group [Formula: see text]. The existence of this symmetry explains the presence of a spectral parameter in the solutions of the equations. We show that similarly, for three-dimensional vertex models and the associated tetrahedron equations, there also exists an infinite discrete group of symmetry. Although generalizing naturally the previous one, it is a much bigger hyperbolic Coxeter group. We indicate how this symmetry can help to resolve the Yang-Baxter equations and their higher-dimensional generalizations and initiate the study of three-dimensional vertex models. These symmetries are naturally represented as birational projective transformations. They may preserve non-trivial algebraic varieties, and lead to proper parametrizations of the models, be they integrable or not. We mention the relation existing between spin models and the Bose-Messner algebras of algebraic combinatorics. Our results also yield the generalization of the condition qn=1 so often mentioned in the theory of quantum groups, when no q parameter is available.

Copper(II) Complexes with cis-Epoxysuccinate Ligand: Syntheses, Crystal Structures, and Magnetic Properties

2011 ◽  
Vol 64 (2) ◽  
pp. 217 ◽  
Author(s):  
Shao-Ming Fang ◽  
E. Carolina Sañudo ◽  
Min Hu ◽  
Qiang Zhang ◽  
Li-Ming Zhou ◽  
...  
Keyword(s):  
Magnetic Properties ◽  
Magnetic Measurements ◽  
Three Dimensional ◽  
Magnetic Coupling ◽  
Supramolecular Interactions ◽  
Stacking Interactions ◽  
Ferromagnetic Coupling ◽  
Two Dimensional ◽  
Structural Comparison ◽  
Higher Dimensional

Three CuII complexes with cis-epoxysuccinate ligand were synthesized and structurally characterized: [Cu(ces)(phen)]2 (1), [Cu(ces)(bpy)]2 (2), and {[Cu2(ces)(pp)2(CH3OH)]}∞ (3), (ces = cis-epoxysuccinate, phen = 1,10-phenanthroline, bpy = 2,2′-bipyridine, and pp = 3-(2-pyridyl)pyrazole with pyrazolyl N-donor deprotonated). Structural analysis reveals that both 1 and 2 have the very similar dinuclear units that are extended by the intermolecular supramolecular interactions, such as C–H⋯O, C–H⋯π, and aromatic π⋯π stacking interactions, to give rise to the higher-dimensional frameworks. Complex 3 has a two-dimensional (2D) layered structure that is further assembled to form a three-dimensional framework by the inter-layer C–H⋯O hydrogen-bonding and C–H⋯π interactions. A structural comparison with those of our previous work in the absence of auxiliary co-ligand suggests that the introduction of 2,2′-bipyridyl-like molecules plays an important role in constructing the final structures of 1–3. Magnetic measurements demonstrate that 1 and 2 exhibit ferromagnetic coupling with the corresponding J values of 1.8 cm–1 for 1 and 1.5 cm–1 for 2, whereas 3 shows more complicated magnetic coupling.

High Resolution Microscopy of Multi-Layered Biological Crystals

1982 ◽  
Vol 40 ◽  
pp. 80-83
Author(s):  
H.A. Cohen ◽  
T.W. Jeng ◽  
W. Chiu
Keyword(s):  
High Resolution ◽  
Low Dose ◽  
Three Dimensional ◽  
Data Interpretation ◽  
Two Dimensional ◽  
Biological Objects ◽  
Density Maps ◽  
Density Map ◽  
Complex Crystal ◽  
High Resolution Microscopy

This tutorial will discuss the methodology of low dose electron diffraction and imaging of crystalline biological objects, the problems of data interpretation for two-dimensional projected density maps of glucose embedded protein crystals, the factors to be considered in combining tilt data from three-dimensional crystals, and finally, the prospects of achieving a high resolution three-dimensional density map of a biological crystal. This methodology will be illustrated using two proteins under investigation in our laboratory, the T4 DNA helix destabilizing protein gp32*I and the crotoxin complex crystal.

Instrumentation For Quantitative Microscopy

1977 ◽  
Vol 35 ◽  
pp. 206-209
Author(s):  
B. Ralph ◽  
A.R. Jones
Keyword(s):  
Volume Fraction ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Automatic Data ◽  
Phase Volume Fraction ◽  
Statistical Confidence ◽  
Resultant Data ◽  
Automatic Data Collection ◽  
Third Dimension ◽  
Evolution Of Microstructure

In all fields of microscopy there is an increasing interest in the quantification of microstructure. This interest may stem from a desire to establish quality control parameters or may have a more fundamental requirement involving the derivation of parameters which partially or completely define the three dimensional nature of the microstructure. This latter categorey of study may arise from an interest in the evolution of microstructure or from a desire to generate detailed property/microstructure relationships. In the more fundamental studies some convolution of two-dimensional data into the third dimension (stereological analysis) will be necessary.In some cases the two-dimensional data may be acquired relatively easily without recourse to automatic data collection and further, it may prove possible to perform the data reduction and analysis relatively easily. In such cases the only recourse to machines may well be in establishing the statistical confidence of the resultant data. Such relatively straightforward studies tend to result from acquiring data on the whole assemblage of features making up the microstructure. In this field data mode, when parameters such as phase volume fraction, mean size etc. are sought, the main case for resorting to automation is in order to perform repetitive analyses since each analysis is relatively easily performed.

A linearity test for thick specimens imaged by HVEM over a 180° angular range

1991 ◽  
Vol 49 ◽  
pp. 552-553
Author(s):  
Yu Liu
Keyword(s):  
Phase Contrast ◽  
Three Dimensional ◽  
Angular Range ◽  
Two Dimensional ◽  
Back Projection ◽  
Line Integral ◽  
Exponential Law ◽  
Transmission Electron ◽  
Absorption Law ◽  
Linearity Test

The image obtained in a transmission electron microscope is the two-dimensional projection of a three-dimensional (3D) object. The 3D reconstruction of the object can be calculated from a series of projections by back-projection, but this algorithm assumes that the image is linearly related to a line integral of the object function. However, there are two kinds of contrast in electron microscopy, scattering and phase contrast, of which only the latter is linear with the optical density (OD) in the micrograph. Therefore the OD can be used as a measure of the projection only for thin specimens where phase contrast dominates the image. For thick specimens, where scattering contrast predominates, an exponential absorption law holds, and a logarithm of OD must be used. However, for large thicknesses, the simple exponential law might break down due to multiple and inelastic scattering.

An Image Reconstruction System Applied to High Voltage Microscopy

1975 ◽  
Vol 33 ◽  
pp. 292-293
Author(s):  
D. E. Johnson
Keyword(s):  
High Voltage ◽  
Prior Information ◽  
Block Diagram ◽  
Three Dimensional ◽  
Real Space ◽  
Two Dimensional ◽  
Efficient Manner ◽  
Fourier Methods ◽  
Tilt Series ◽  
Methods Of Reconstruction

Increased specimen penetration; the principle advantage of high voltage microscopy, is accompanied by an increased need to utilize information on three dimensional specimen structure available in the form of two dimensional projections (i.e. micrographs). We are engaged in a program to develop methods which allow the maximum use of information contained in a through tilt series of micrographs to determine three dimensional speciman structure.In general, we are dealing with structures lacking in symmetry and with projections available from only a limited span of angles (±60°). For these reasons, we must make maximum use of any prior information available about the specimen. To do this in the most efficient manner, we have concentrated on iterative, real space methods rather than Fourier methods of reconstruction. The particular iterative algorithm we have developed is given in detail in ref. 3. A block diagram of the complete reconstruction system is shown in fig. 1.

Computer Assisted Image Reconstruction of Oligomer Structure

1976 ◽  
Vol 34 ◽  
pp. 472-473
Author(s):  
A.M. Jones ◽  
A. Max Fiskin
Keyword(s):  
Three Dimensional ◽  
Depth Of Field ◽  
Computer Assisted ◽  
Projection Data ◽  
Two Dimensional ◽  
Single Particles ◽  
Dimensional Reconstruction ◽  
Computer Assisted Image ◽  
The Fourier Transform ◽  
Space Approach

If the tilt of a specimen can be varied either by the strategy of observing identical particles orientated randomly or by use of a eucentric goniometer stage, three dimensional reconstruction procedures are available (l). If the specimens, such as small protein aggregates, lack periodicity, direct space methods compete favorably in ease of implementation with reconstruction by the Fourier (transform) space approach (2). Regardless of method, reconstruction is possible because useful specimen thicknesses are always much less than the depth of field in an electron microscope. Thus electron images record the amount of stain in columns of the object normal to the recording plates. For single particles, practical considerations dictate that the specimen be tilted precisely about a single axis. In so doing a reconstructed image is achieved serially from two-dimensional sections which in turn are generated by a series of back-to-front lines of projection data.

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